Conversely, let \(\rho \)  be a separable state.Clearly, \(\rho ^{T_\mathcal {A}}\)  is positive, equivalently \(\phi _{\rho ^{T_\mathcal {A}}}\)  is completely positive in this case.
Then \(\phi _\rho \)  is an entanglement-breaking channel [1]}, i.e., there exist states \(E_k \in \mathcal {S}(\mathcal {B})\)  and positive operators \(F_k \ge 0\)  such that
\(\phi _\rho (a)= \sum _{k=1}^K E_k\ \mathrm {tr}_{\mathcal {H}_\mathcal {A}}[F_k a]\; .\) 
